nt.number theory - Classification of strongly lcm-closed sets

I call a set X of positive integers strongly lcm-closed if a,b ∈ X if and only if lcm(a,b) ∈ X. In the finite case X is the set of divisors of lcm_{x ∈ X}x. But in the infinite case it is more interesting, for example, ${a geq 1: a notequiv 0 pmod p}$ and ${p^a:a geq 0}$ for any prime p, are strongly lcm-closed sets.

Which sets are strongly lcm-closed sets?

This question arose in my Ph.D. thesis (p.107) where strongly lcm-closed sets describe where autotopisms of Latin squares give rise to subsquares.

As a side question:

Is there a common name for strongly lcm-closed sets?

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